{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p15 - Solve eigenvalue BVP $u_{xx} = \\lambda u$, $u(-1)=u(1)=0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from numpy import dot,argsort,linspace,shape,zeros,polyval,polyfit,pi,real\n",
    "from chebPy import cheb\n",
    "from scipy.linalg import solve,eig\n",
    "from matplotlib.pyplot import figure,subplot,plot,title,axis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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   "source": [
    "N = 36\n",
    "D,x = cheb(N)\n",
    "D2 = dot(D,D)\n",
    "D2 = D2[1:N,1:N]\n",
    "\n",
    "lam,V = eig(D2)   \n",
    "ii = argsort(-lam)\n",
    "lam = real(lam[ii])\n",
    "V = V[:,ii]\n",
    "\n",
    "fig = figure(figsize=(10,15))\n",
    "for j in range(5,35,5):              \n",
    "    lv = shape(V)[0]+2\n",
    "    u = zeros(lv)\n",
    "    u[1:lv-1] = V[:,int(j)]  \n",
    "    subplot(6,1,j/5)\n",
    "    plot(x,u,'bo')\n",
    "    xx = linspace(-1.0,1.0,201)\n",
    "    uu = polyval(polyfit(x,u,N),xx)    # interpolate grid data\n",
    "    s = 'eig %d = %20.13f * 4/$\\pi^2$' %(j,lam[j-1]*4/pi**2)\n",
    "    s = s + '\\t\\t %4.1f ppw' % (4*N/(pi*j))\n",
    "    title(s)\n",
    "    plot(xx,uu,'b')\n",
    "    axis('off')"
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